The ell=3 framing anomaly and Kirby blow-up

The labels are $0,1$. The dimension formula gives $d_0=d_1=1$. The twist formula gives ${\theta }_0=1$ and ${\theta }_1=q^{3\mathrm{/}2}=e^{\pi i\mathrm{/}2}=i$. Thus $D^2=1+1=2$ and choose $D=\sqrt{2}$. The Gauss sums are $p_{+}=1+i,p_{-}=1-i,\kappa =\frac{1+i}{\sqrt{2}}=e^{\pi i\mathrm{/}4}\mathrm{.}$ In particular $\kappa \mathrm{\ne }1$.

A positive blow-up multiplies the surgery link value by $p_{+}$ and changes $(m,\sigma )$ to $(m+1,\sigma +1)$, so its normalized ratio is $D^{-1}{\kappa }^{-1}{p}_{+}=D^{-1}\frac{D}{p_{+}}{p}_{+}=1.$ A negative blow-up has ratio $D^{-1}\kappa p_{-}=D^{-1}\frac{p_{+}{p}_{-}}{D}=1$ by [F2]. If the factor ${\kappa }^{-\sigma }$ is omitted, the same two blow-ups multiply the value by $p_{+}\mathrm{/}D=\kappa$ and $p_{-}\mathrm{/}D={\kappa }^{-1}$, both nontrivial. Thus an unweighted surgery link sum is not invariant under these moves for this actual category; this is a failure of omitting the signature correction, not an assertion that the weight-zero arrows form a subcategory.

The actual modular quotient is supplied by [A1], so the preceding values are realized category data. The full weighted functor uses ${\kappa }^{n+a-\sigma (B)}$ together with its exact outgoing cap factors. A Kirby blow-up changes neither the marked bordism, its weight nor its cap-reference correction, and step 2.1 cancels its complete surgery ratio. A unit increase in bordism weight instead multiplies the map by $\kappa =e^{\pi i\mathrm{/}4}\mathrm{\ne }1$. Thus the actual category has a nontrivial framing anomaly, and the proved weighted functor includes it with the positive weight convention.